risksensitive planning support for forest enterprises the yafo model.pdf

8/11/2019 Risksensitive planning support for forest enterprises The YAFO model.pdf

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Risk-sensitive planning support for forest enterprises: The YAFO model

Fabian Hrtl , Andreas Hahn, Thomas Knoke

Institute of Forest Management, Center of Life and Food Sciences Weihenstephan, Technische Universitt Mnchen (TUM), Hans-Carl-von-Carlowitz-Platz 2, 85354 Freising, Germany

a r t i c l e i n f o

Article history:

Received 4 October 2012

Received in revised form 11 March 2013

Accepted 14 March 2013

Keywords:

Economic optimization

Risk integration

Operational planning

Forest management planning

Nonlinear programming

Long-term objectives

a b s t r a c t

YAFO is a planning-support tool for the development of management plans under uncertainty focusing on

the forest enterprise level. Based on existing stand data, the software provides the calculation of manage-

ment scenarios (felling plans) for single stands that are optimized with respect to financial considerationsand ecologica l constraints. Under these constraints, YAFO predicts timber stocks, harvest amounts and

financial returns for each simulation period. The YAFO package consists of an optimization module, that

has been programmed using the modell ing software AIMMS. In addition, it contains two Excel-based

spreadsheet files an import and evaluation module and a risk analysis module. The YAFO model calcu-

lates financially optimized management scenarios by means of the net present value development of sin-

gle stands. Optionally, the objective function can also consider risk s and uncertainties due to natural

calamities and timber price fluctuations, using the value at risk approach or risk utility functions. Non-

linear programming algorithms are used as solution techniques. As YAFO provides the additional flexibil-

ity to switch between two timber grading options on stand level, effects of timber price scenarios on

grading can be analyzed. Due to its modular design, it can be easily adopted to individual data bases.

2013 Elsevier B.V. All rights reserved.

1. Introduction

If one is to approach the problem of managing a forest enter-

prise in a sustainable1 way, it is necessary to tackle the question

of when to harvest which timber volume from which stand (forest

area with the same treatment). Due to the long production periods

in forests, especially in Central Europe, this decision is crucial in or-

der to avoid negative consequences that can potentially last for dec-

ades. It is then no surprise that there is a long tradition of planning

techniques in forestry to address this problem. Georg Ludwig Hartig

(Hartig, 1795) and Heinrich Cotta (Cotta, 1804) are generally consid-

ered to be the first forest scientists to have developed such applica-

ble solution techniques as regulation by forest area and harvested

volume respectively. These techniques are commonly known as

control techniques in forestry literature (Davis et al., 2001; Bettin-

ger, 2009). In the English forestry literature, there has been a contin-uous enhancement of these initial forest planning techniques,

culminating in the integration of methods from decision theory,

operations research and finance theory into forest enterprise man-

agement (Davis et al., 2001; Buongiorno and Gilless, 2003; Rauscher,

2005; Reynolds et al., 2008).Thus, planning/decision support systems (DSSs) correspond to

specific eras of forest management, starting with sustained yield,

and finally emerging in sustainable forest management (SFM)

(Mathey et al., 2005; Hahn and Knoke, 2010). Mendoza (2005) dif-

ferentiates two approaches for decision support in forestry one

prescriptive, algorithmic and highly structured, and the other

descriptive, soft and qualitative. He states, the latter has become

more popular and more widely applied, in part because of its affin-

ity to the participatory management approach (Mendoza, 2005, p.

252). Participatory decision making, and ecologically and socially

sound decisions are primarily related to intragenerational fairness

a cornerstone of SFM. Intergenerational fairness, however the

second cornerstone of the World Commission on Environment

and Developments (WCED, 1987) definition of a sustainable devel-opment, and the originally relevant criteria for sustainable forestry

(Hahn and Knoke, 2010) is less frequently addressed. Timber har-

vests are thus a matter of allocation where cuttings have to be car-

ried out in an efficient way with regard to future harvesting options.

Hence, our interest focuses on the producers perspective, as weas-

sume sustainable forest management to be best promoted if land-

owners personally benefit. At these scales, research activities in

recent years have developed spatially explicit and geographically

sensitive systems (Varma et al., 2000; Reynolds et al., 2008). A

second point of action led to an increased emphasis on the adapta-

tion options due to serious changes in the decision environment

(Eriksson, 2006; Heinimann, 2010; Mermet and Farcy, 2011).

0168-1699/$ - see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.compag.2013.03.004

Corresponding author. Tel.: +49 8161 71 4619; fax: +49 8161 71 4545.

E-mail address: [email protected] (F. Hrtl).

URL: http://www.waldinventur.wzw.tum.de(F. Hrtl).1 Following Speidel (1984) sustain able here means the abilit y of a forest

enterprise to provide timber, infrastructure and additional goods and services for

the benefit of present and future generations in a continuous and optimal manner

(Knoke et al., 2012). For definition problems concerning this freque ntly used term

refer for example to Hahn and Knok e (2010).

Computers and Electronics in Agriculture 94 (2013) 5870

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In general, the classic forest planning problem of allocating

areas to space and time can be considered as maximising an objec-

tive function. If one considers the single stands of a forest enter-

prise as options of a finance portfolio, one can build an objective

function that calculates the net present value (NPV) of all manage-

ment activities during the planning horizon. By means of linear

programming (LP) methods, it is possible to solve this optimization

problem by computer-assisted numerical algorithms (Felbermeieret al., 2007), which covers the highly structured and algorithmic

approach Mendoza, 2005 figured out.

LP is by far the most used method in forest planning as well as

in general land-use optimization (Bettinger and Chung, 2004;

Weintraub and Romero, 2006). Current research focuses on exten-

sions to LP like mixed integer (Fonseca et al., 2012) and goal (mul-

ti-objective) programming (Diaz-Balteiro and Romero, 2008; Rivaz

and Yaghoobi, 2012), on non-linear approaches (Hof and Kent,

1990; Roise, 1990), on heuristic (stochastic) approaches like simu-

lated annealing (Georgiou and Papamichail, 2008), tabu search and

genetic algorithms (Mosquera et al., 2011; Janov, 2012; Pukkala

and Kellomki, 2012), and on dynamic programming (Benjamin

et al., 2009), as well as on combining these techniques with spa-

tially explicit models (Seppelt and Voinov, 2002; Baskent and

Keles, 2005; Gustafson et al., 2006; Mathey et al., 2008; Wei and

Murray, 2012) and risk considerations (Martell et al., 1998; Kangas

and Kangas, 2004; Knoke et al., 2005; Mathey and Nelson, 2010;

Verderame et al., 2010).

Several papers (e. g. Gong, 1998; Knoke et al., 2001; Knoke and

Moog, 2005; Alvarez and Koskela, 2006; Beinhofer, 2009; Roessiger

et al., 2011) have shown that forest management decisions are se-

verely influenced by risks. Such uncertainties can be caused by tim-

ber price fluctuations (Brazee and Mendelsohn, 1988; Haight, 1990)

as well as calamities (Meilby et al., 2001; Kouba, 2002; Hahn and

Knoke, 2010; Forsell et al., 2011; Hanewinkel et al., 2011; Griess

et al., 2012). So systems need to incorporate these deviations in or-

der to set up sustainable solutions. The evaluation of these uncer-

tainties can be accomplished for example through Monte Carlo

simulation techniques (Styblo Beder, 1995; Dieter, 2001). If theaim is to integrate such risk effects into a model, a potential solu-

tion is to describe the resulting fluctuations of financial returns by

statistical values, such as mean and standard deviation (mean-var-

iance analysis). This statistical approach was used by Markowitz

(1952, 1959) in his portfolio theory (Mills and Hoover, 1982; Hilde-

brandt and Knoke, 2011). This approach assumes a Gaussian distri-

bution of the fluctuating net revenues, but it has been shown to be

robust to deviations from normality (e. g. Glawischnig and Seidl,

2011). Other approaches for overcoming this limitation are repre-

sented by models based on stochastic dominance, downside risk,

and information gap theory (Knoke et al., 2008). Due to their math-

ematical structure, models considering risk effects in general must

be treated as nonlinear in finding a solution (Pukkala and Kangas,

1996; Knoke and Moog, 2005; Hildebrandt and Knoke, 2009;Knoke et al., 2012). Nevertheless there are approaches to the inclu-

sion of uncertainties in linear programming techniques using ma-

trix models (Eriksson, 2006) or in stochastic integer programming

(Alonso-Ayuso et al., 2011), or different measurements of risk, such

as absolute deviations (Konno and Yamazaki, 1991).

Literature is, however, largely missing approaches which allow

an easy and quick parameterisation of this risk-sensitive planning

problem with abdication of assumptions concerning linearity

(Yousefpour et al., 2012). Many research projects about DSS are

based on forest growth simulators to which are added capabilities

to optimize the planning with regard to biophysical objectives. A

few examples are LMS/FVS (McCarter et al., 1998; Crookston and

Dixon, 2005), SAGALP (Chen and Gadow, 2002), HEUREKA (Lmas

and Eriksson, 2003), SADfLOR (Borges et al., 2003), DSD (Lexeret al., 2005), MOTTI (Salminen et al., 2005), NED-2 (Twery et al.,

2005), FTM (Andersson et al., 2005), HARVEST (Gustafson and Ras-

mussen, 2002), 4S TOOL (Kirilenko et al., 2007), AFFOREST (Gil-

liams et al., 2005), EMDS (Reynolds, 2006), ESC (Pyatt et al.,

2001), FSOS (Liu et al., 2000), and FORESTAR (Shao et al., 2005).

Other solutions like SIMO (Rasinmki et al., 2009) or Woodstock

(Remsoft Inc., 2012) try to go beyond biophysical objectives but

are acting more as a model development tool than a model itself.

Furthermore, modelling approaches integrating risks, like the FOR-EST OPTIMIZER project (Stang and Knoke, 2009) are scarce, and

also retain a linearisation of risks. For an overview of different ap-

proaches see Bjrndal et al., 2012

We therefore see the need for a further development of an algo-

rithmic approach to address the question of optimal risk-sensitive

management on the forest enterprise level over time using nonlin-

ear programming (NLP). Thus, the model presented here is aimed

at making a planning and decision tool available to forest scien-

tists, as well as practitioners, that can be used to solve a multitude

of problems without requiring any major adaptions.

The model considers not only the risk effects mentioned above,

but also the effects of different timber price scenarios. Climate

change mitigation policies as well as a fear of increasing scarcity

of fossil fuels provokes a growing demand for producing energy

from biomass. Due to that increase, the prices of fuel wood are ris-

ing, so that the competition between the material and thermal use

of wood is becoming more and more severe (Raunikar et al., 2010).

To analyse the effects of these competing lines of timber use, we

expand the model to include an option to decide simultaneously be-

tween two timber grading options during the allocation of stand

areas.2 The combination of risk analysis with Monte Carlo simula-

tions, grading options and NLP techniques is a new way to handle

the planning problem at the enterprise level. For that purpose, the

model generates probability distributions of the objective function

out of the original data, using timber price statistics and survival

functions for tree species. Finally, this combination of risk analysis

on enterprise level with Monte Carlo techniques and NLP is unique

so far and not available in the packages mentioned above.

In all of the model approaches mentioned here, it is possible toanalyse effects of constraint settings which simulate demands for

maintaining or providing ecological or social functions of forests.

Comparing constrained solutions for the objective with uncon-

strained ones gives us the opportunity to evaluate the costs of such

ecosystem services (Duraiappah, 2005), for example, how much

money a forest owner requires in exchange for providing such

functions. In this way we solve the problem of non-existent mar-

kets and prices for such ecosystem functions, at least from a pro-

viders perspective (Knoke et al., 2008).

2. Method

2.1. Basic model

YAFO3 is a modular nonlinear optimization model for forest

enterprises. It is based on a forest property that is spatially divided

into forest stands. Every stand i is an independent management unit

that is characterized, from a financial point of view, through the

development over time of its net revenues. The stands cannot split

up or merge within the model. The model consists of seven time

periods numbered from 0 to 6 the last of which is a recovery per-

iod that collects all remaining stand areas at the end of the investi-

gated time horizon. No thinning or felling is carried out in the last

period, instead, the remaining area of the stands not felled during

the simulated time horizon is stored which is then used as a factor

2

This is optional. The model can also be used for one-scenario optimizations.3 Yet Another Forest Optimi zer.

F. Hrtl et al./ Computers and Electronics in Agriculture 94 (2013) 5870 59


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in calculating the net present value of the stands. The remaining six

periods span a time horizon of 3060 years, as typical time steps in

forest growth models are 5 or 10 years. At every point in time t a

decision must be made to either thin or finally fell parts of the stand

area. To thin means that the stand remains, at least until the next

time period, and that only single trees will be removed for stand

improvement. The cutting intensity of these thinnings is normally

defined by silvicultural concepts, and is not decided by the model.Thinnings produce intermediate returns during the rotation period.

To fell means to cut the entire stand, or parts of it, at the end of

the rotation period and establish a new stand generation. Regenera-

tion costs must then be paid. These costs are determined based on

the dominant tree species as well as the stand age. The older a stand,

the lower the regeneration costs, to simulate possibilities for natural

regeneration. This cost reduction is done by a regeneration cost

moderation function that follows a Weibull function, and can be ad-

justed to various local situations by its parameters. For planned as

well as salvage fellings a new stand generation is simulated with

its ingrowth volumes for the following periods. These ingrowth vol-

umes are included in the calculation of thinning and felling volumes

only from period five onwards as it is assumed that there will be no

utilizable ingrowth volumes in stands younger than 25 years.

Furthermore, the model decides in each case which grading op-

tion is to be applied to that particular harvest. The model is free to

choose between these options. Additionally, in every period, a cer-

tain partial area,fzits , of each stand must be cut (salvage felling). This

mechanism simulates expected tree drop-outs caused by wind,

snow or insects, and is based on a hazard rate that is calculated

as a function of the leading tree species, the mixture conditions,

and the stand age (see Section 2.2.2). At every point in time, t,

the model decides for every stand, i, in addition to the determined

salvage returns,zits, whether it is more profitable to realise the re-

turns of an (intermediate) thinning, dits, or those of a (final) felling,

aits, as a portfolio option. The index, s, shows that the model must

also choose between two grading options for every individual

stand in every period, which have different revenues and costs.

The optimizer can realize the thinning data (volumes, revenuesand costs) by assigning the stand areas or parts of it to be thinned

to the variables fdits. The remainder of the stand area is then as-

signed to the variablesfaits that realizes the final felling of the resid-

ual stand (volumes, revenues and costs).

The sum of the net present values (NPV) of all these net reve-

nues is the objective function that is to be optimized by the area

control method. The objective function has therefore the following

form:

maxf

ZXi

Xt

Xs

ditsfdits aitsf

aits zitsf

zits

1 r

t 1

with the constraints,

Xs

fdit0s

Xt0

t0

Xs

faits fzits

fi 8i; t

0 2a

Xs

fzits fzit 8i; t 2b

fd;a;zits P 0 8i; t; s 2c

The meaning of the symbols is as follows: rinterest rate, ttime, i

stand, s grading option,fi area of stand i, dits revenues per area from

thinning (net-of harvesting costs) in stand i at time tusing grading

option s,aits revenues per area from felling (net-of harvesting costs),zits revenues per area from salvage felling (net-of harvesting costs),

fdits thinning area, faits felling area, f

zits area of salvage felling. Con-

straint (2a) assures that for every point in time, t0

, the sum of the

area felled to date plus the current area to be thinned is equal tothe stand area. This means that every area not yet felled is thinned

automatically. Constraint (2b) ensures that the salvage felling area

in each period cannot be used as a thinning or final felling option.

Constraint (2c) prohibits solutions with negative areas.

This area allocation problem itself is modelled as an area control

scheme that allows stand areas to be shifted in space and time

using the modelling software AIMMS (Paragon Decision Technol-

ogy B.V., 2011). For every timber grading option there exists a sep-

arate scheme. The combination of both schemes is accomplishedusing the constraints according to Eqs. (2a) and (2b). This approach

has the advantage that model and data are strictly separated, so

that it is quite simple to use data sets that do not rely upon the data

preparation and evaluation module YAFO-EX. AIMMS symbolizes

the model in a tree structure. All model components are placed

in this tree as single objects. The objective functions as well as

the optimization problems are placed as objects in the model.

The former are categorized as variables in the AIMMS language,

the latter as mathematical programs. The connections between

these objects are implemented through object declarations.

The area for each stand in all periods is defined by seven con-

straints. This set of constraints represents the side condition

according to Eq. (2a) in the model as follows. For each period t

there is the following constraint:Xs

fdits faits

fRit 8i; t 3

with the recursive defined remaining area

fRitn :fRitn1

Xs

faitn1sfzits

4

After expanding the recursion the right side of Eq. (4) can be

combined in a different way:

fRitn fRit0

Xn1x0

Xs

faitxsXnx1

Xs

fzitxs

fifzit0

Xn1x0

Xs

faitxsXnx1

Xs

fzitxs

fiXn1x0

Xs

faitxsXnx0

Xs

fzitxs 5

Relinquishing the counting index x for the different points in

time tin Eq. (5) leads to the simplified formulation

fRit0 fiXt01t0

Xs

faits Xt0

t0

Xs

fzits 6

Substituting Eq. (3) into Eq. (6) gives

Xs

fdit0sfait0s

fi

Xt01t0

Xs

faits Xt0

t0

Xs

fzits

and finally after rearrangement the structure of Eq. (2a):

Xs

fdit0s

fiXt0

t0

Xs

faits Xt0

t0

Xs

fzits fiXt0

t0

Xs

faits fzits

7

So Eq. (3) with (4) and Eq. (2a) are identical.

Six additional constraints represent the salvage felling area con-

trol of Eq. (2b) for each period 05. The non-negativity constraint

(2c) is incorporated directly into the variable declarations. Addi-

tionally the following five biophysical constraints can be defined

at the enterprise level:

Lower limit of standing volume in (m3/ha)

Upper limit of standing volume in (m3/ha)

Maximum final felling volume in (m3/ha/period)

Maximum final felling area in (ha/period) Maximum total felling volume in (m3/ha/period)

60 F. Hrtl et al./ Computers and Electronics in Agriculture 94 (2013) 5870


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Again, six constraints are defined for each of these enterprise le-

vel parameters, separately for each period 05, to force the vari-

ables to the interval between the constraints. Period 6, as the

recovery basket, is affected only by the first two constraints. For

this purpose the model updates and saves the biophysical develop-

ment of the stands. With the exception of the enterprise-level con-

straints, the biophysical data do not affect the optimization

process. The main function of the biophysical data is to give theuser additional facts for management planning and to check the re-

sults. Carrying biophysical data as well as financial data through

the model system enables the calculation of a timber amount that

follows the optimized planning solution for the forest enterprise.

The actual realised thinning volumes are calculated by multiplying

the growth model-based thinning volumes by the thinning areas.

Similarly, the felling volumes are calculated by multiplying the

simulated stand volume by the felling area. The volume of the

stand after thinning is computed by multiplying the simulated

stand volume by the difference between the total stand area and

area already used.

The spreadsheet calculation YAFO-EX prepares the data so that it

can be imported bythe optimizer model YAFO-A (Fig. 1). Section 3.1

describes the data YAFO-EX requires as an input. The stands are

assigned to one of four categories representing hardwood and soft-

woods in pure and mixed stands. Using survival functions, age

dependent drop-out partial areas due to calamities are calculated

for every stand in each period. The timber volume is classified for

further analysis by the main tree groups spruce, pine, beech and

oak, and by the main classes saw log, industrial wood, fuel wood

(from compact wood) and brushwood. The brushwood amounts

are reduced by a exploitation factor which is chosen by the user in order to calculate economically usable amounts. Regeneration

costs provided by a separate data sheet are added to each stand.

Finally, in order to calculate the NPVs, an interest rate must be

entered.

The data processing described above is controlled by the user

through buttons provided in a central control sheet (Fig. 2). The

buttons are associated with VBA codes that activate the pro-

grammed commands, so that the handling is quite straightforward.

Through a series of processing steps, the data set is rearranged to

match the matrix format used by the area control scheme in the

model, with the stands in rows, and the periods in columns. The

name manager of Excel is used to assign name spaces to the data.

The I/O interface of YAFO-A accesses these names to assign the

model parameters to the appropriate data as listed in Section 3.1

Fig. 1. Flowchart of YAFO-EX and YAFO-MC (dashed box).



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(see also Fig. 3). The central control sheet also provides additional

buttons to activate the Monte Carlo simulation, YAFO-MC, that cal-

culates the spreading of the potential NPV of each stand in each

period by simulating the variations described in Section 2.2.2,

and derives variation coefficents and correlation matrices. After it

is computed, the risk data is written back to YAFO-EX (see

Fig. 1). Section 3.1 gives a list of the data the solution contains.

In addition to performing these preparatory work, YAFO-EX

evaluates the optimized results (see Fig. 1). For that purpose,

YAFO-A uses the I/O interface to export its solution back to

YAFO-EX. Based on this, YAFO-EX provides summaries illustratingthe progress of the stock, the harvest volumes, the NPVs, the felling

values, value increments and area distribution of the stand devel-

opment classes.

2.2. Risk integration

2.2.1. Value at risk and risk utility

To incorporate the risk effects already mentioned, the objective

function (1) must be expanded. For this purpose, certainty equiva-

lents that are derived from utility functions (Gerber and Pafumi,

1998; Bamberg et al., 2008) can be used, for example. Alternatively,

minimum values according to the Maximin decision rule can be

optimized (Young, 1998; Hildebrandt and Knoke, 2009; Hilde-

brandt and Knoke, 2011). As the worst case scenario for a givenobjective is normally very unlikely and therefore can be considered

irrelevant, or as is the case of a continuous distribution function

the probability of this worst case approaches zero, it makes sense

to focus on a defined threshold that is exceeded with a given prob-

ability (Mowrer, 2000). Such a limit is nothing other than a certain

quantile of the risk-driven probability function of the objective. In

finance this concept is well known as the value at risk (VAR)

(Stambaugh, 1996; Jorion, 1997; Knoke et al., 2012). If the realisa-

ble net revenues d, a andzfrom thinnings and (salvage) fellings are

distributed by risk effects and interpreted as expected values with

statistical spreads, the expected value of the objectiveZis distrib-

uted as well. FZrepresents the distribution function of that risk-dri-

ven objective function Z. The related inverse function F1Z p then

defines the p-quantile ofFZ, the value, that is exceeded byZwitha probability of 1p. The new objective is as follows:

maxf

Z F1Z p 8

Thus, the objective no longer optimizes the uncertain expected

value ofZbut instead, the worst value forZthat can be expected

with a certain probability of 1p. The assumption of a Gaussian

distribution

Z N EZ; s2Z

FZ 9

defines this distribution by the expected value E(Z) and the variance

s2Z. Using this precondition, F1 can be calculated as the inverse of a

normal distribution.Another approach to handling risk effects is the use of utility

functions that reduce the expected return with a weighted vari-

ance according to the assumed risk aversion of the manager. In this

case the objective function can be written as a certainty

equivalent:

maxf

Z EZ a

2s2Z 10

Herea is a constant, representing the absolute risk aversion of thedecision-maker.

If the decision-maker behaves as a risk-seeker, this can be easily

modelled with both approaches. In the case of Eq. (8) it is possible

to optimise other p-quantiles. A p-quantile of 0.5 represents a risk-

neutral behavior, whereas p-quantiles above 0.5 correspond torisk-seeking management decisions. In Eq. (10) a negative a canbe used to simulate a risk-seeking behavior.

2.2.2. Risk simulation by Monte Carlo

The two parameters E(Z) and s2Z, that are needed for the deter-

mination of the distribution function can be estimated, for exam-

ple, from local experience. But the model presented uses a

different approach: The parameters are estimated based on real

data by use of the integrated Monte Carlo (MC) module, YAFO-

MC, prior to the optimization process. The Monte Carlo simulation

is implemented as Visual Basic (VBA) code in a separate Excel file

that is linked to YAFO-EX. For both grading options, a separate

MC module is provided. The modules generate, by default, 1,000

possible proceeds and costs for every stand and period each cal-culated as a NPV sum of discounted net revenues from the possible

Fig. 2. Screenshot of the YAFO-EX central control sheet.



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final felling in that single period and the thinnings and salvage fel-

lings done thus far, by randomly modifying the evaluated growth

simulator data (see Fig. 1) by a mechanism for timber price fluctu-

ation and another one for calamity occurence. Using the nomencla-

ture ofPritsker (1997), this approach is afull Monte Carlo method,

as each draw is based on exact pricing.Timber price fluctuations are treated as a random variable. For

each simulation step there is a randomly chosen year between

1975 and 2010 that is associated with that step. That year numberdefines factors that weight the returns in this period using simple

multiplication. There are two different factors for hard and soft-

wood. These factors are calculated from the timber price statistics

published for the Bavarian state forest, and denote the percent

deviation of that years timber prices from the average price during

the time horizon mentioned. All prices are adjusted for inflation.

The prices are derived from two chief timber grades average

quality spruce timber, diameter class 2529 cm, for softwood,

and average quality beech, diameter class 4049 cm, for hardwood.

Inflation is estimated based on the so-called Long Series of the

German consumer price index (DESTATIS, 2011).

To randomize the appearance of a calamity this aspect is mod-

elled here in a different way as in the optimizer model YAFO-A.

Not average ratios of salvage areas per period are used but a ran-dom number between 0 and 1 is picked for every stand. In each

period one calamity is possible in every stand, and there can be just

one calamity for each stand during the simulated time horizon.

This random number is compared to the hazard rate computed

for the single stand. If the random number exceeds the hazard rate,

the calamity occurs. The stand is then felled as a whole and a new

stand generation is planted. The hazard rate is calculated using

survival functions according to Griess et al. (2012). The change of

the survival function st etb

a

during a given period in time h,

with respect to the initial state, defines the hazard rate a(t):

at st st h

st h 11

Different empirically derived values for a andb are determinedfor each of four stand types pure and mixed softwood stands as

well as pure and mixed hardwood stands. Returns due to calami-

ties are reduced with a calamity factor which is chosen by the user.

The associated salvage fellings are considered prior to regular fel-

lings in each period.

2.2.3. Risk evaluation

The 1000 simulation runs generate 1000 possible NPVs for

every stand in each period and each grading option. These NPVs

are saved for each period. From this data, a correlation matrix iscalculated between the uncertain NPVs as well as a variation coef-

Fig. 3. Flowchart of YAFO-A.



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ficient for each stand. It is possible to do this either at the stand le-

vel or to combine the stands to groups for this purpose. In the latter

case, the distribution of the average values for each group is calcu-

lated. To do this, YAFO-A requires a group attribute for each stand

(see Section 3.1). As the standard option in YAFO-EX, nine groups

are used (spruce, fir, pine, larch, douglas fir, beech, oak, valuable

hardwood, other hardwood).

Let bgts represent a vector containing the possible realisations ofthe uncertain NPV in period tand grading option s for the defined

stand group g= 1,2, . . . ,n. Then every matrix Kts, with periods

t= 0,1,. . . ,6, contains the correlations between these vectors:

Kts :

corrb1ts;b1ts corrb1ts;bnts

.

.

...

...

.

corrbnts;b1ts corrbnts;bnts

0BB@

1CCA 12

The covariances Vxyts are calculated for each period tand grad-

ing option s by multiplying the correlation coefficient Kxyts between

two stands or groupsx undy by the areasf(d,a,z), used by the model

inx undy, by the variation coefficients v of the NPVs, and by the

NPVs d0t: dt(1 + r)t, a0t: at(1 + r)

t and z0t:zt(1 + r)t that

can be realised in the stands or groups,x andy. Thus, the covari-ance matricesVts have the following components:

Vxyts : Kxyts vxts vyts fdxtsd0xtsf

axtsa0xtsf

zxtsz0xts

fdytsd0ytsfaytsa0ytsf

zytsz0yts

13

These covariance matrices are summed up by elements to cal-

culate the total variance

s2ZXx;y;t;s

Vxyts 14

of the objective functionZ. The variance of the last period is divided

by five to account for the fact that the model algorithm cannot dis-

tribute its decisions in this period forward into the future as can be

done in reality, because the model does not cover future periods.

The model can spread felling areas from period six to five or four,

although the particular stand might not have reached the NPV peak.

Taking the full variance of period six into the model causes an over-

estimated felling area in the preceding period, whereas reducing the

variance to zero lets the model try to avoid the fellings and to reach

the risk-free period six. The parameterisation of this factor must

balance these two opposing decisions in a reasonable way.4

The expected value ofZ, defined in Eq. (1), and the variance just

calculated define the distribution function FZ, as shown in Eq. (9).

The inverse function F1Z p to be maximised according to Eq. (8),

is also defined. In the model, this function is not calculated as

the inverse ofFZ but instead, a reduction factor is used. Assuming

a Gaussian distribution according to Eq. (9), the difference between

the expected value of the objective function EZ F1Z 0:5 andthe value at risk F1Z p can be expressed in terms of multiples q

of the standard deviation sZ ofZ. This multiplication factor, that

is equivalent to the desired value at risk quantile p, is equal to

the quantile q of the standardised normal distributionU(q), so that

U(q) = 1p. Therefore, the objective function of Eq. (8) can be cal-

culated by

F1Z p EZ qsZ: 15

Knoke and Wurm (2006) have shown that the spreading of re-

turns from forests follows a Gaussian distribution only in an

imperfect manner. Therefore, the optimizing model presented here

uses the Monte Carlo simulation mentioned to generate a more

realistic spreading of the uncertainty factors as a first step. In the

following nonlinear objective function, this approach is then sim-

plified by describingthis simulated spreading like a Gaussian one.According to Beinhofer (2009), the quality of the predicted results

is not highly affected through this simplification, as long as confi-

dence levels 1p below 95% are used.

So, three optimization programs exist in YAFO-A (Fig. 3): A sim-

ple NPV maximisation (Eq. (1)) and two programs considering risk:

value at risk VaR (Eq. (15)) and certainty equivalent CE (Eq. (10)).

All three problems are classified in AIMMS as nonlinear, although

the NPV maximisation is actually linear. This is due to the fact that

AIMMS does not distinguish between variables5 that are part of the

chosen objective function and those which are not. For all three

cases, AIMMS uses CONOPT (Consulting and Development A/S, xxxx;

Drud, 1994), a solver algorithm for nonlinear programs that search

for a local optimum. Consequently, we define subsets of variables

and constraints to construct a mathematical program that is ableto solve the NPV maximisation as a linear problem. Using these sub-

sets, a simplex algorithm can be applied to determine a global opti-

mum. The algorithm used in this case is the ILOG CPLEX solver (IBM

Corp., 2011).

The model also contains procedures (program code) that help

the user to automatize the process of solution finding. These proce-

dures can be initiated through buttons on the user interface

(Fig. 4). There are three main procedures that carry out the three

mathematical programs described above. In addition, to find a glo-

bal optimum of the nonlinear problems, there is also a multi-start

option. In multi-start mode, YAFO-A uses the multi-start module

included in AIMMS to search for an optimum, starting from the

20 best solutions that are calculated by 100 randomly selected

starting points. This search is repeated 10 times. For further detailsabout the multi-start module see Rinnooy Kan and Timmer (1987)

and Roelofs and Bisschop (2011).

3. Application and example

3.1. Data preparation

The users of our model must prepare their data in an Excel or

Calc sheet. To do so, the modular design of the model offers two

options. The users have the option of using the evaluation tool

YAFO-EX. This tool is designed to read stand data produced by for-

est growth simulators, simulate uncertainties with the help of the

coupled Monte Carlo module YAFO-MC, and deliverthese data sets

to the optimizer model. Alternatively, they can use a data manipu-lation of their own, as done by Hahn et al. (submitted for publica-

tion), and import the data directly to the optimizer YAFO-A via the

defined I/O Interface.

The spreadsheet file YAFO-EX uses a single sheet for each of the

two possible grading options, that must provide for each stand of

the investigated enterprise the following data structure line by

line:

Stand identifier/number

Year or period

An identification for thinning data (year or period value) and

residual stand data (0)

Proceeds and cost per hectare (/ha)

4 A spruce dominated stand in Bavaria typically reaches the maximum NPV in

between 60 and 100 years. Forty years or eight periods are necessary to cover this

period. Therefore, seen from the point of period six, there are, eight future periods

missing in the model to determine the correc t point in time when the NPVs of the

stands existing in period six, are reaching the maximum. On average, eight of nine

stands in period six are not mature. The redu ction of the variance in period sixprevents the model to spread these non-mature stand areas into period five. 5 In AIMMS every object that can be changed in the model is a variable.



8/13

Total volume of (compact) wood (m3/ha)

Volume of brushwood (m3/ha) (optional) Dominant tree species

Stand age (years)

Wood volume of each timber grade class (m3/ha)

This data set is required for each period for the thinning and the

residual stand. For seven points in time, this implies 14 rows for

every stand in the data sheet. By entering additional data for

brushwood, the tool is able to calculate the capabilities for provid-

ing fuel wood amounts from brushwood. In order for these

amounts to be considered financially, they must be included in

the expected proceeds and costs per ha.

The I/O interface of YAFO-A imports data from any Excel or Calc

sheet, and exports the solution as well. To ensure data is assigned

to the correct model objects, the cell areas in Excel or Calc must bemarked with defined names. YAFO-EX already provides these

name conventions. The following data is imported by YAFO-A:

Stand identifier/number

Initial area of each stand (ha)

Initial age of each stand (years)

List of groups for stands grouping

Assignment of each stand to the groups

Thinning volumes for each stand, period and grading option

(m3/ha)

Stock volumes for each stand, period and grading option (m3/

ha)

NPV of proceeds and costs for each stand, period and grading

option from thinnings, fellings and salvage fellings (

/ha) Hazard rate of each stand in each period (%)

Correlation matrix for each period

Variation coefficents for each stand/group in each period

The following values are exported to the target solution file

(YAFO-EX as standard):

NPV sum ()

Value at risk ()

Certainty equivalent ()

Lists of areas used in each period and grading option by thin-

ning, salvage and final felling (ha)

Covariance matrix for each period

3.2. Example

As an example we demonstrate the application of YAFO on adata set of inventory plots of the second German federal forest

inventory BWI 2 (BMVEL, 2005) that has been projected by the

growth simulator WEHAM (Bsch, 2004a,b). For this purpose, the

growth simulator was set so that there was a possibility to thin

the stands but not to fell them finally, as the timing of the final fell-

ing will be determined by our model. To calculate brushwood

amounts we used volume expansion factors according to Zell

(2008). The data set tested consists of 267 plots (satellite sample

plots as used in the inventory) that belong to the state forest of

the geographical region Tertires Hgelland in Bavaria. These

plots are considered as 267 stands of a forest enterprise, each rep-

resenting a stand of 1 ha. The data show a softwood-dominated

tree species distribution with high standing timber volumes

(410 m3

/ha) that are typical for this region. One hundred fifty-sixof these 267 ha are covered by spruce-dominated stands, and

Fig. 4. Graphical user interface (GUI) of YAFO-A.



9/13

52 ha are covered by beech. The other stands are dominated by

pine or other hardwoods. The growth simulator WEHAM has an

integrated grading function, so that, without the use of any addi-

tional programs, we obtain a graded result of the biophysical

development of the stands. We compare two grading variants:

The first scenario is meant to represent the actual grading practices

used in forestry at present, and emphasizes the material use of tim-

ber, with only moderate fuel wood amounts from small-sizedwood. The second scenario emphasizes the thermal use of wood,

by having minimum diameters for saw log and industrial wood

that are 12 cm larger than those used in the first scenario. Using

a database of our own, these volumes are evaluated and trans-

formed to the data structure YAFO-EX requires. The assumed tim-

ber prices and costs of harvesting are shown in Table 1. The

harvesting costs for spruce are used for all softwood species, and

those for beech for all hardwoods. Fir prices are set 5 below,

and pine prices 20 below spruce. The prices for larch and douglas

fir are set 10 above spruce. Beech prices are used for all hard-

woods except oak and low value hardwoods. Prices for oak are gi-

ven in the table. Low value hardwoods are priced at 5 below

beech.

The assumed regeneration costs are shown in Table 2. The

regeneration costs modification is implemented as a Weibull func-

tion following the form etb

a

where tis the stand age and the two

parameters are defined as a : 70 and b: 5. The data used byYAFO-EX to simulate volume and thinning amounts from ingrowth

are given in Table 3.

The factor to reduce net revenues from calamities as well as the

brushwood exploitation factor are set to 0.5, the interest rate to 2%,

and the value at risk quantile to 5%. We do not introduce any fur-

ther constraints at the forest enterprise level. The risk simulation

through the Monte Carlo module is complete after approximately

10 min.

We optimize the 267-stand enterprise in YAFO-A with and

without risk aspects. The linear program gives a global optimum

of 17,081 /ha for the NPV. The nonlinear optimization with risk ef-

fects is calculated using the multi-start option, resulting in thesolution of 14,690 /ha for the VAR (NPV at 17,001 /ha). The sum-

mary results for the timber production of the model enterprise are

shown in Tables 4 and 5.

The data in the tables are aggregated to the enterprise level and

displays logging volumes, area development and financial results.

This data can be used for supporting the decisions of the forest

manager. It is possible to retrace this data to the single stands.

So an operational felling plan in terms of a stand list can be pro-

vided for the manager.

4. Discussion

4.1. Model

The main focus of YAFO is the economic analysis of felling sce-

narios when making decisions based primarily on financial values.

Many other approaches are also capable of calculating financial

values, but either do not provide the possibility to consider them

as decision variables (for example LMS/FVS, DSD, FTM, 4S TOOL,

AFFOREST), or do not integrate all risk aspects financial as well

as natural ones (for example HEUREKA, SIMO, DSD, NED-2).

YAFO, however, provides the consideration of (biophysical) restric-

tions in the solution process, so that it is possible to implement

ecological and social barriers, at least to the extent that they can

be expressed using such constraints. For example, in order to main-

tain a certain level of ecosystem services (e. g. recreation, water

conservation) there can be the additional objective to maintain aspecific minimal average timber volume within the forest enter-

prise. To do so, it is easily possible to formulate a final timber vol-

ume that must be remain at the end of the investigated

management period.

In contrast to most other approaches, YAFO does not use linear

programming or heuristic algorithms to solve the decision prob-

lem, but rather NLP techniques. The advantage of this method over

linear models is that the risk aspects mentioned above can be eas-

ily integrated. The uncertainties included in the YAFO model coverrisks due to timber price fluctuations, as well as calamity probabil-

ities, and their relationship to species mixture. Unlike further ex-

tended frameworks of uncertainty (Williams, 2012), the model

assumes that the objectives are known and accepted. In NLP it is

possible to determine an optimal solution by using solver algo-

rithms for global optima, or as in our case by calculating it with

the help of advanced multi-start techniques. In contrast, heuristic

approaches can achieve only approximate solutions.

The spatially implicit nature of the model is achieved through

the consideration of the stands of the forest enterprise in the area

control scheme (Turner et al., 2001; Perry and Enright, 2007). This

approach fulfills level 4 of spatial recognition, as defined by Davis

et al. (2001). Further spatial effects, such as direct interactions be-

tween stands are not considered, as the YAFO model does not in-

clude information about the spatial arrangement of the stands, as

for example, SAGALP or HARVEST do.

The YAFO model is not designed as a monolithic solution with

an integrated growth simulator. Instead it is a modular tool to sup-

port managers of private and public forest land in their decisions.

YAFO can be easily adapted to existing growth simulators, due to

its open data interface. According to Reynolds (2005), it is argu-

able if optimization systems are real DSS (Rauscher et al., 2007).

YAFO certainly can be considered as a DSS component, based on

the definition by Holsapple (2003) of DSS as problem-processing

systems supporting a decision-making process. According to the

definition given in Menzel et al. (2012), the tool can be recognized

either as a typical part of a DSS, or as a DSS in a wider sense. They

searched for criteria to merge the quantitative, analytical with the

qualitative, more participatory oriented approach, as differentiatedby Reynolds et al. (2008). YAFO as a new, innovative, and algorith-

mic model complies with the eight criteria Menzel et al. (2012) de-

rived for evaluating a DSS from a participatory planning

perspective.

One potential weakness of the modular approach is the lack of

interaction between optimization and growth simulation. The

growth prediction is not able to act in response to the thinning ac-

tions planned by the optimization. Thus, the thinnings used in the

financial model are determined by and limited to the decisions

made by the growth simulation. The provision of such interactions

is still a big advantage of the monolithic approach of other systems,

such as HEUREKA. However, due to the lack of a globally valid for-

est growth simulation, the ability of YAFO to use data from various

growth simulators represents a big advantage of the modular ap-proach. Existing growth simulators are parameterized for specific

regions. For example, SILVA has implemented growth functions

mainly for Bavaria, BWINPro for North West Germany, DSD for

southern Austria, LMS/FVS for the United States, SADfLOR for Por-

tugal, and HEUREKA for Sweden. Combining a universally valid

financial model with a specific local/regional growth simulator

solution would inhibit a broader use of our system, as such all-

in-one solutions are usually not easily adaptable to other regional

(growth) conditions. Our target is to provide an open tool that can

be linked to different growth models. Although this design means a

higher workload for the users, as they must take care to appropri-

ate data import and export from one program to the other, in the

end it provides greater flexibility (see e.g. Nute et al., 2005). The

example above shows that interaction with the WEHAM growthsimulator.



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Another advantage to the YAFO model is its speed. The risk sim-

ulation through the Monte Carlo module is complete after approx-

imately 10 min, and the calculation time of YAFO-A is only about

2 min.6 For two grading options, at six points in time across 267

stands, in which the model distinguishes between thinning and fell-

ing actions as well as grading options for the salvage fellings, the

optimization problem consists of 9612 independent decision vari-

ables and 6974 constraints.

7

In total then, the model must calculateabout 39,000 variables. Our previous attempts using the Excel Add-

in, Whats Best (LINDO Systems Inc., 2011), for nonlinear program-

ming required several hours of computation time and did not reach

feasible solutions in either case. The redesign of the model within

AIMMS is therefore a big step towards practicality.

4.2. Example

Analysing the solution for the risk-ignoring linear NPV optimi-

zation one can see that the high initial average stand volume of

410 m3/ha is immediately reduced by a heavy harvesting operation

of 19 m3/ha/a distributed across 33 ha of the enterprise area. This

decreases the timber increment to 11 m3/ha/a. The main reason

for this is the large percentage of high volume spruce stands in

age classes IV (6079 years) and V (8099 years) that have already

reached, or even exceeded, their financially optimal rotation age.

The average volume is reduced during the following periods down

to 304 m3/ha in period five. From this point it increases over the

next 5 years, ending at 349 m3/ha. A little bit more than one third

of the forest area (99 ha) is felled during the considered time

horizon.

In contrast, the harvests undertaken by the risk-sensitive vari-

ant are more equally distributed. The initial harvest amount is only

12 m3/ha/a, whereas the harvest volume in the subsequent periods

ranges between 11 and 14 m3/ha/a. In these periods, the risk-

ignoring variant harvested between 9 and 12 m3/ha/a. The total fi-

nal felling area of 90 ha is below the one in the first case, while the

minimum average stand volume rises to 320 m3/ha, finally ending

at 367 m3/ha in period 6. The main reason for these results is thatthe algorithm tries to arrange the harvesting activities in a more

evenly distributed fashion to avoid high risk effects. This is quite

clear in the development of the area in spruce of age class V. The

risk-ignoring variant reduces this age class through its first harvest

action from 39 ha to 26 ha. In contrast, the value at risk optimiza-

tion distributes this reduction across four consecutive harvests, not

reaching a comparable level of 28 ha remaining spruce until period

three 15 years later.

This equalizing tendency also affects the financial results. While

the risk-free optimization allows a drop in periodical revenues

from an initial 689/ha/a to 461 /ha/a including major fluctua-

tions, the results of the value at risk variant are more continuous:

The maximum of 433/ha/a in period zero is accompanied by a fi-

nal value of 533/ha/a. Although the revenues reduce to 376 /ha/a in period two, this minimum exceeds the minimum of the NPV

variant (294 /ha/a) by 82/ha/a. The results thus show that deal-

ing with risks in a forest enterprise planning process leads to a

more equally distributed logging plan, and therefore allows the

owner to benefit from a more balanced flow of net revenues.

Another result can be seen by analysing how the timber har-

vested is split up between the two grading options. In both optimi-

zation approaches, about 10% of the timber is graded according to

the fuel wood scenario. This timber comes from the younger hard-

wood stands, because it is more profitable to grade these small-

sized hardwoods as fuel wood and sell them, for example, via con-

tract felling than for the owner to conduct the harvest and sellthem as saw logs (refer to the assumed price scenario in Table 1).

Table 1

Income revenue over harvesting cost for spruce and beech.

Species Diameter

class (cm)

Avg. price

(/m3)

Harvesting costs

(/m3)

Income revenue

(/m3)

Spruce 614 40 21 19

1519 47 22 25

2024 59 20 39

2529 64 19 45

3034 64 18 463539 61 17 44

4044 60 16 44

P45 60 17 43

Industrial

wood

40 21 19

Fuel wood 10 0 10

Beech 614 40 26 14

1519 42 25 17

2024 44 22 22

2529 44 21 23

3034 49 19 30

3539 62 18 44

4044 72 16 56

4549 72 16 56

5054 80 18 62

P55 84 18 66

Industrialwood

40 26 14

Fuel wood 20 0 20

Oak 614 40 26 14

1519 40 25 15

2024 40 22 18

2529 58 21 37

3034 76 19 57

3539 103 18 85

4044 140 16 124

4549 140 16 124

5054 164 18 146

P55 174 18 156

Industrial

wood

40 26 14

Fuel wood 20 0 20

Table 2

Regeneration costs.

Tree species Costs (/ha)

Beech 6400

Valuable hardwood 5225

Other hardwood 4895

Oak 8250

Spruce 1600

Fir 2700

Douglas fir 3958

Pine 3630

Larch 1400

Table 3

Young stand data of thinning and volume growth.

Tree species Age (years) Thinning (m3) Volume (m3)

Softwood 20 15 40

25 20 60

30 30 100

Hardwood 20 0 10

25 5 25

30 10 40

6 We used a PC with an Intel Core i5-2400 CPU and 3.1 GHz.7 7 267 for every stand, plus 6 267 for the salvage felling, plus non-negativity

constraints of the same size, plus 32 optional biophysical constraints at the enterpriselevel.



11/13

Having the flexibility to grade simultaneously in two ways is one of

the key strengths of the model. Of particular advantage in our ap-

proach is the opportunity to compare two grading variants not

just by means of their total NPV or value at risk but also to be

able to analyse such gradual shifts between the grading options

at the individual stand level.

5. Summary

The goal of the development of the YAFO optimization model is

to provide a planning-support tool that can be easily adopted for

risk neutral, risk averse and risk-seeking decision makers, as well

as to a wide variety of problems. Therefore it sets a high valueon the strict division between model representation and data man-

agement. We use commercially available software that can usually

be acquired with low costs at least for the research sector. The

modular concept including defined interfaces between the mod-

ules allows for the combination of the users data in different

ways within the model: Either a graded and valuated data set via

the Excel tool YAFO-EX (for example, for processed data from

growth simulators), or a set of coefficients with separate risk eval-

uation (correlation matrices and variation coefficients) for directimport into the optimizer, YAFO-A, can be used. The use of the

Monte Carlo module, YAFO-MC, is optional, as the system can han-

dle nonlinear problems with uncertainty evaluations as well as

simple linear problems. The model can also distinguish between

two grading options. The version of the tool presented here is inde-

pendent of the number of stands that need to be investigated. One

current limitation is the number of periods in time that can be con-

sidered. The ability to find local or global optima for nonlinear

problems is not a question of the presented model but of the solver

the users apply in their AIMMS system. As AIMMS provides an

open interface for that reason, it is possible to combine our model

with different solver algorithms.

The example shown performs as expected: Both the periodical

biophysical and financial results occur more smoothly when the

optimization considers uncertainties due to risk effects. Another

interesting result is the response to the two different grading op-

tions. As described, this decision is made by the model at the indi-

vidual stand level. In the future, it might be helpful to examine this

effect in greater detail. Thus, the model may enable us to analyse

the behaviour of forest owners in response to various timber price

scenarios. For example, it would be valuable to investigate how

timber amounts shift between the material and thermal-use tim-

ber grades under the assumption of increasing prices for fuel wood

that are likely given the expected continued increase in oil prices.

Acknowledgements

The study presented here is part of the Project G33 Competi-tion for wood: Ecological, social and economic effects of the mate-

rial and energetic utilization of wood funded by the Bavarian State

Ministry of Food, Agriculture and Forestry, and as Project

22009411 by the German Federal Ministry of Food, Agriculture

and Consumer Protection. The authors wish to thank Laura Carlson

and Yolanda Wiersma for the language editing of the manuscript

and two anonymous reviewers for valuable suggestions.

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Table 4

Results of the forest enterprise optimization without considering uncertainties.

Period 0 1 2 3 4 5 6 Avg.

Initial stock (m3/ha) 410 372 376 370 378 366 349 375

Remaining stock (m3/ha) 317 319 317 327 319 304 349 322

Regular felling (m3/ha/a) 18 10 11 8 11 12 0 12

Salvage felling (m3/ha/a) 1 1 1 1 1 1 0 1

Total felling (m3/ha/a) 19 11 12 9 12 12 0 12

Increment (m3/ha/a) 11 11 11 10 9 9 10

Areas (ha)

Intermediate felling 234 220 205 197 184 168

Final felling 29 12 13 7 12 14

Salvage felling 4 2 2 2 2 2

Regeneration 33 47 62 70 83

Sum 267 267 267 267 267 267

Logging volume (m3/ha/a)

Saw log 15 8 9 6 9 9 9

Industrial wood 2 2 2 2 2 2 2

Firewood 2 1 1 1 1 1 1

Brush firewood 2 1 1 1 1 1 1

Profit (/ha/a)


Final felling 473 217 260 115 248 263

Sum regular felling 676 362 420 288 434 452

Salvage felling 13 7 6 6 7 8Sum 689 368 426 294 441 461

Table 5

Results of the forest enterprise optimization maximising the value at risk.

Period 0 1 2 3 4 5 6 Avg.

Initial stock (m3/ha) 411 408 405 407 407 390 367 399

Remaining stock (m3/ha) 350 345 350 353 341 320 367 347

Regular felling (m3/ha/a) 12 12 10 10 12 13 0 11

Salvage felling (m3/ha/a) 1 1 1 1 1 1 0 1

Total felling (m3/ha/a) 12 13 11 11 13 14 0 12

Increment (m3/ha/a) 12 12 11 11 10 9 11

Areas (ha)

Intermediate felling 254 236 224 211 195 177Final felling 9 14 10 10 13 15


Regeneration 13 31 43 56 72

Sum 267 267 267 267 267 267

Logging volume (m3/ha/a)

Saw log 9 10 8 8 10 11 9

Industrial wood 2 2 2 2 2 2 2

Firewood 1 1 1 1 1 1 1

Brush firewood 1 1 1 1 1 1 1

Profit (/ha/a)


Final felling 210 278 196 186 282 323

Sum regular felling 420 430 368 374 483 523


Sum 433 441 376 383 491 533



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